Consistency Results on the Structure of Measures on a Measurable Cardinal

Measurable cardinals and normal measures were introduced by Ulam and have since become central objects in modern set theory. A landmark theorem of Friedman and Magidor studies the possible number of normal measures that a measurable cardinal may carry. In this talk we will present new related results. The first is a new and simpler proof of the Friedman-Magidor theorem, which allows new applications that the original theorem did not have. The second, based on joint work with Ben-Neria, establishes the consistency of Goldberg’s Ultrapower Axiom with the failure of GCH at a measurable cardinal. The first half of the talk will introduce the basic definitions and lead up to the main results, and will be accessible to a broad mathematical audience. In the second half we will sketch the new proof of the Friedman-Magidor theorem, assuming familiarity with forcing.